A Full-frontal Hero Arises in the World!

A Full-frontal Hero Arises in the World! is the 1st episode of the first season of Heaven's Lost Property.

Summary
At the beginning, we see Tomoki, as he explains his recurring dream. In his dream he meets an Angel (Daedalus), but does not know her name yet, and he's in love with her. It soon shifts to a scene usually seen before the episode begins. In the next scene we first meet Eishirou Sugata who explains more about his research. In this episode, he presents his "assistant" Pretty, and explains about his research concerning the mysterious "black hole" hovering over the Earth.

Later Sohara tries to wake up Tomoki, who is dreaming, and she made the mistake of moving the covers and shrieks seeing his "morning wood". He tries to make an excuse about it but was then punished by Sohara (via karate chop). Back at school, he sleeps during class and encounters the Angel (Daedalus) again. Sohara wakes him up, and he had a sad look along with tears in his eyes. Sohara showed some sympathy to him and tried to convince him to see a doctor, but he shrugs off saying it's nothing.

Nether the less, Sohara wants to bring him to Sugata, because of Tomoki's dreams to ask for advice, as his dreams seem to cause him suffering. Suddenly the whole class was in front of the entrance to the school, and Sugata has positioned itself with his glider on the roof. This is also the first time we meet Mikako, the school president, who is apparently insured up to 10 million yen for Sugata. After he jumped, he flew for a few seconds before he crashes into a tree. Because of this he had received a few wounds, but they weren't serious. After that Sugata and the gang meet up in his lab where Sohara explains about reluctant Tomoki's dreams. Sugata asserts that he knew what the dream meant, and he theorized it was about the "New World". He connects it to that of the black hole and believes its a portal to the "New World". Sugata arranges to meet at midnight at the cherry blossom tree to see if the black hole would appear. However, Sugata was stopped by Mikako (for unknown reasons), and Sohara couldn't go out after midnight. It was just Tomoki waiting there alone. He then receives an emergency call to get the hell out of there. But it was too late.

The hole is directly above him, and an object falls directly in front of him, blowing him back. The strange object created a crater, and Tomoki looks into the crater to see that is is not an object, but a girl clad in black in white, pink hair, and big wings. After some hesitation, he pulls her out of the rubble. When trying to bring her out of the danger zone, he is almost killed by a pillar. He screams and looks away, but the creature spread her wings and flew away from the pillar and up into the sky. He opened and eyes to think he is flying, but turns to face the angel. She then does the Imprinting Process, embodied by the chain wrapped around Tomoki's hand and connect to her neck collar. She lands and reveals who she was: a Pet-Class Angeloid. The next day, Tomoki wakes up to see her, and she offers him to give her a command. This resulted with Tomoki becoming rich, which was made possible by Ikaros. He also altered time so he could have some fun, at least for him (such as running around naked with a spear, looking at girl's underwear, etc.) After Tomoki accidentally wishes Ikaros to strip in front of him, he unknowingly wishes for world domination. The next morning Tomoki wakes up to see everyone is gone. Ikaros explains that when he wished to become a world ruler, no one would see him as their king, so everyone was erased because of the card.

Throughout the day, he became depressed, as he couldn't decided whether to be sad or angry. Ikaros apologizes and asked if she would commit suicide. Tomoki didn't take this seriously and said yes, saying he had enough trouble. Just before Ikaros would sadly shoot herself, Tomoki tackles and stops her, begging her to abort the command and wished that everything would have been just a dream. He begs her to stay with him, as he would be alone otherwise. And so the day begins again, but as a dream. Tomoki sees Ikaros again, and she asked if she was to be apart of his dream. Tomoki says no, and wants her to do what she wants. She didn't understand, but she understood to stay by his side

Tomoki's Dream
Tomoki stated that he was in love with the girl in his dream. In his second dream Daedalus told him to treasure the Angeloid, and stay with her. This possibly means she wants Tomoki to be with Ikaros instead of her.

Trivia

 * Tomoki's first dream of Daedalus is very likely not a dream at all, but is rather the fragmented memories of the time when Daedalus was acting as the original Sohara before her death and reawakening in Synapse as shown at the end of the series.